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Use the notation of L144. Throughout, $0\le a\le2^{-1/3}$, $c=\sqrt{1-4a^6}$, and

$$P(z)=z^3+3a^2z,\qquad q_a=P-c,\qquad L(a)=\ell(q_a),\qquad L_0=L(0)=\ell(z^3-1).$$

Put $\beta=\arcsin(2a^3)\in[0,\pi/2]$, so that $c=\cos\beta$.

1. Reduction of all cubics to the family

By Lemmas 4 and 5 of Eremenko and Hayman, some monic cubic $p^*$ maximizes $\ell$ and has all its critical points on its lemniscate, so its critical values have modulus one. A translation and the rotation $p(z)\mapsto e^{-3i\theta}p(e^{i\theta}z)$ preserve monicity and length, so we may take $p^*(z)=z^3-3u^2z+b$ with $b\le0$. Its critical values are $b\mp2u^3$, and $|b-2u^3|=|b+2u^3|=1$ gives $\operatorname{Re}(b\bar u^3)=0$ and $b^2+4|u|^6=1$. If $b<0$, then $u^3$ is purely imaginary, and a rotation by a multiple of $2\pi/3$, which fixes $b$, makes $u=\pm ia$ with $a=|u|$. If $b=0$, an arbitrary rotation achieves the same normalization. Thus $p^*=q_a$. This is the normalization of the research sketch. Hence every monic cubic $p$ has $\ell(p)\le\ell(p^*)=L(a)$ for some $a$, and the inequality for every cubic follows from the family inequality.

2. The length integral

The lemniscate of $q_a$ is $P^{-1}(C)$, where $C=\{s:|s-c|=1\}$. The critical points of $P$ are $\pm ia$, with critical values $s_\pm=\pm2ia^3$, and both lie on $C$ since $4a^6+c^2=1$. Away from these two values, $P$ restricted to $P^{-1}(C)$ is a three-sheeted covering by analytic arcs, and on each arc $|dz|=|ds|/|P'(z)|$. Therefore

$$L(a)=\int_0^{2\pi}G_a(c+e^{i\theta})\,d\theta,\qquad G_a(s)=\sum_{P(z)=s}\frac1{|P'(z)|},$$

and the conjugation symmetry $G_a(\bar s)=G_a(s)$ gives $L(a)=2\int_0^\pi G_a(c+e^{i\theta})\,d\theta$. On $C$ the critical values sit at $\theta_\pm=\pi\mp\beta$. For $a=0$ the roots of $z^3=s$ give $G_0(s)=|s|^{-2/3}$.

3. Root identities

If $P(z)=s$, expansion gives

$$(z^2+a^2)^2(z^2+4a^2)=z^6+6a^2z^4+9a^4z^2+4a^6=s^2+4a^6.$$

Since $P'(z)=3(z^2+a^2)$, for $s\ne s_\pm$,

$$\frac1{|P'(z)|}=\frac{|z^2+4a^2|^{1/2}}{3|s^2+4a^6|^{1/2}}.$$

Let $U$ be either root of $U-a^6/U=s$ and let $u_1,u_2,u_3$ be the cube roots of $U$. Then $(u-a^2/u)^3+3a^2(u-a^2/u)=u^3-a^6/u^3=s$, the roots of $P(z)=s$ are $u_j-a^2/u_j$ (the other choice of $U$ is $-a^6/U$, whose cube roots are $-a^2/u_j$, giving the same roots), and $z^2+4a^2=(u+a^2/u)^2$. Hence

$$G_a(s)=\frac{\sum_j|u_j+a^2/u_j|}{3|s^2+4a^6|^{1/2}}. \tag{1}$$

For $a>0$ put $\sigma=s/a^3$. Then $G_a(s)=a^{-2}G_1(\sigma)$, where $G_1$ is (1) with $a=1$.

4. The comparison with $z^3-c$

Let $\Phi(c)=\int_0^{2\pi}|c+e^{i\theta}|^{-2/3}\,d\theta$, the length of $\{|z^3-c|=1\}$. For $0\le c<1$, $(1+ce^{i\theta})^{-1/3}=\sum_kb_kc^ke^{ik\theta}$ with $b_k=\binom{-1/3}k$, and Parseval gives $\Phi(c)=2\pi\sum_kb_k^2c^{2k}$, strictly increasing for $c>0$. To pass to $c=1$, use $|c+e^{i\theta}|^2=(1-c)^2+c|1+e^{i\theta}|^2$ for $c\ge1/2$ and dominated convergence; the limiting singularity has exponent $-2/3$. Monotone convergence of the series and the beta integral at $c=1$ give

$$\Phi(c)\le\Phi(1)=L_0=2\pi\,{}_2F_1\!\left(\tfrac13,\tfrac13;1;1\right) =\frac{2\pi\,\Gamma(1/3)}{\Gamma(2/3)^2} =2^{1/3}B\!\left(\tfrac12,\tfrac16\right)\approx9.1797.$$

Indeed $|1+e^{i\theta}|=2|\cos(\theta/2)|$ evaluates the integral as the beta value displayed, and the gamma duplication formula gives the gamma quotient.

For $a>0$ write $g(\sigma)=G_1(\sigma)-|\sigma|^{-2/3}$ and let $\Gamma_a$ be the circle $|a^3\sigma-c|=1$, the image of $C$. Since $G_a(s)=a^{-2}G_1(\sigma)$, $|s|^{-2/3}=a^{-2}|\sigma|^{-2/3}$ and $|ds|=a^3|d\sigma|$,

$$L(a)=\Phi(c)+a\,I(a),\qquad I(a)=\int_{\Gamma_a}g(\sigma)\,|d\sigma|, \qquad\text{so}\qquad L(a)\le L_0+a\,I(a). \tag{2}$$

5. Small parameters: $0<a\le13/64$

Here it suffices that $I(a)<0$. The circle $\Gamma_a$ has center $c/a^3$ and radius $a^{-3}$ and passes through $\pm2i$. For $a\le13/64$ one has $a^6\cdot50^2<1$ and $c/a^3>50$. Hence the arc of $\Gamma_a$ near the origin is the graph

$$x=x_a(y)=\frac{a^3(y^2-4)}{c+\sqrt{1-a^6y^2}},\qquad |y|\le50,\qquad |d\sigma|=\frac{dy}{\sqrt{1-a^6y^2}},$$

and every other point of $\Gamma_a$ has $|\sigma|\ge50$. The integrand and the arc are symmetric under conjugation.

Tail. Let $|\sigma|\ge Y>2$, let $T$ be the root of $T-1/T=\sigma$ with $|T|\ge1$, so $|T|\ge|\sigma|-1$, let $\tau$ be a cube root of $T$, $\varepsilon=\tau^{-2}$ and $e=|\varepsilon|=|T|^{-2/3}\le(Y-1)^{-2/3}$. From (1) with $a=1$,

$$g(\sigma)=|T|^{-2/3}\left(\frac{A(\varepsilon)}{|1+\varepsilon^3|} -|1-\varepsilon^3|^{-2/3}\right),\qquad A(\varepsilon)=\frac13\sum_{j=0}^2|1+\varepsilon\omega^{-2j}|,$$

with $\omega=e^{2\pi i/3}$. The real parts of the three terms of $A$ sum to three, and $|1+w|\le1+\operatorname{Re}w+|w|^2/(2(1-|w|))$, so $1\le A\le1+e^2/(2(1-e))$. With $(1+e^3)^{-2/3}\ge1-\tfrac23e^3$ and $(1-e^3)^{-2/3}\le(1-e^3)^{-1}$ this gives $|g(\sigma)|\le K|T|^{-2}\le K(|\sigma|-1)^{-2}$, where

$$K=\max\left\{\frac{e+1/(2(1-e))}{1-e^3}+\frac{2e}3,\ \frac{2e}{1-e^3}\right\}, \qquad e=(Y-1)^{-2/3}.$$

Parametrize $C$ by $s=c-e^{i\psi}$. Denote the near arc by $\mathcal A_Y$; it is $|\psi|\le\psi_Y$ with $\sin\psi_Y=a^3Y$, and on the rest $|s|\ge a^3Y$ and $|s|^2\ge4c\sin^2(\psi/2)$. Hence

$$\int_{\Gamma_a\setminus\mathcal A_Y}\frac{|d\sigma|}{(|\sigma|-1)^2} \le\frac{a^3}{(1-1/Y)^2}\int_{\psi_Y<|\psi|\le\pi}\frac{d\psi}{|s|^2} \le\frac{a^3\cot(\psi_Y/2)}{c(1-1/Y)^2}\le\frac{2}{cY(1-1/Y)^2}.$$

With $Y=50$ this tail is at most $0.03$ in absolute value.

Near arc. For each box $a\in[k/64,(k+1)/64]$, $0\le k\le12$, the program bounds $2\int_0^{50}g(x_a(y)+iy)(1-a^6y^2)^{-1/2}dy$ from above, uniformly for $a$ in the box. On $[0,2^{-10}]$ it uses $|\sigma|\le X+y$, with $X$ the largest $|x_a|$ there, so the subtracted term is at least $3[(X+2^{-10})^{1/3}-X^{1/3}]$. On $[2-2^{-13},2+2^{-13}]$ it uses $G_1(\sigma)=N_1(\sigma)/(3|\sigma-2i|^{1/2}|\sigma+2i|^{1/2})$ with a ball enclosure of the numerator and $|\sigma-2i|\ge|y-2|$, so that the window contributes at most the enclosure times $4\cdot2^{-13/2}$. Elsewhere it sums the supremum of a ball enclosure of the integrand over each piece times the piece width. Every box gives $I(a)<0$; the largest bound is about $-0.216$. By (2), $L(a)<L_0$ for $0<a\le13/64$.

The first parameter box includes $a=0$ only as an enclosure of the continuous formula for $x_a(y)$ and its weight. The circle $\Gamma_a$ and (2) are used only for $a>0$; at $a=0$ the target equality is exact. The head estimate integrates the $|y|^{-2/3}$ singularity rather than evaluating it at zero, and the window estimate integrates the $|y-2|^{-1/2}$ singularity. Thus no positive lower cutoff on $a$ or on distance to either singularity is assumed.

6. Larger parameters: $7/512\le\beta\le\pi/2$

Here the program bounds $L(a)=2\int_0^\pi G_a(c+e^{i\theta})d\theta$ itself, uniformly for $\beta$ in each box, with $c=\cos\beta$ and $a^3=\sin\beta/2$. A window $W$ contains $\theta_+=\pi-\beta$ for every $\beta$ in the box. On it, $G_a(s)=N_a(s)/(3|s-s_+|^{1/2}|s-s_-|^{1/2})$ by (1), with $|s-s_+|=2|\sin((\theta-\theta_+)/2)|\ge(2/\pi)|\theta-\theta_+|$ and $\int_W|\theta-\theta_+|^{-1/2}d\theta\le2\sqrt{2|W|}$; so the window is bounded by the supremum of $N_a/(3|s-s_-|^{1/2})$ over $W$ times $(\pi/2)^{1/2}\,2\sqrt{2|W|}$. The point $\theta_-=\pi+\beta$ lies outside $[0,\pi]$. Off the window the program sums piece suprema of ball enclosures of (1). Starting from 300 equal boxes and bisecting any box whose bound is not below $L_0-1/100$, it covers $[7/512,\pi/2]$ by 591 consecutive boxes, each with $L(a)<L_0-1/100$.

The program checks $0<\inf W<\sup W<\pi$ and inclusion of every possible $\theta_+$ before accepting a window. It integrates the nonnegative density up to an upper dyadic enclosure $\pi_+\ge\pi$ for an upper bound. The parameter cover likewise ends at a dyadic enclosure of $\pi/2$ from above, so it includes $c=0$ without assuming $c>0$. The density formula remains valid in this small extension by taking $c=\cos\beta$ and $a^3=\sin\beta/2$.

7. Coverage and conclusion

The program checks $7/512<\arcsin(2(13/64)^3)$, so the two regimes overlap and together cover $0<a\le2^{-1/3}$. Hence $L(a)<L_0$ for every $a$ in that range. With Section 1, every monic cubic $p$ satisfies $\ell(p)\le L_0$.

On the nonsingular pieces, every bound in Sections 5 and 6 is a finite sum of exact piece widths times enclosing endpoints of arb balls. The explicit head, window and tail estimates bound all omitted pieces. All formulas take the whole parameter box as their input, so the bound holds at every parameter in that box, rather than only at sampled values. Stopping an integration subdivision at its depth limit retains the same certified rectangle bound; a nonfinite enclosure fails. The adaptive tolerances affect only sharpness.

The square-root and cube-root choices in (1) cause no loss of coverage: either quadratic root and any cube root give the same multiset of three roots of $P(z)=s$. A fixed branch is enclosed on each input ball; the code rotates its cut when convenient. No holomorphic integration assumption is used. As a control, the program also computes a lower bound at $\beta=1/2$ using lower ends on the regular pieces, omitting the singular window, and stopping at a lower dyadic enclosure $\pi_-\le\pi$. Nonnegativity makes these omissions valid. The two bounds are $8.1746\le L\le8.1830$.

8. Uniqueness of the maximizer

Let $p$ be a monic cubic of maximal length; then $\ell(p)\ge L_0$. Translate so that $p(z)=z^3-3u^2z+b$.

If $u=0$, Section 2 with $G_0(s)=|s|^{-2/3}$ on the circle $|s+b|=1$ gives $\ell(p)=\Phi(|b|)$ when $|b|\le1$ and $\ell(p)=|b|^{-2/3}\Phi(1/|b|)$ when $|b|>1$. Since $b_1=-1/3\ne0$, $\Phi$ is strictly increasing, so $\ell(p)<L_0$ unless $|b|=1$.

If $u\ne0$, the critical points $\pm u$ are simple and the critical values $b\mp2u^3$ are distinct. If both lie on the unit circle $\mathbf T$, Section 1 makes $p$ a rotation and translation of $q_{|u|}$, and Sections 5 to 7 give $\ell(p)<L_0$, a contradiction. Otherwise use the deformation in the proof of Eremenko and Hayman's Lemma 5, which applies to every extremal polynomial: for a critical value $v\notin\mathbf T$ and an open disc $U\ni v$ missing $\mathbf T$ and the other critical values, it gives monic polynomials $p_\lambda=\phi_\lambda\circ p\circ\psi_\lambda^{-1}$, $|\lambda|<\epsilon$, with $\phi_\lambda,\psi_\lambda$ homeomorphisms, the critical value $v$ moved to $v+\lambda$, the other critical value fixed, and $\ell(p_\lambda)=\ell(p)$. Each $p_\lambda$ again has two simple critical points.

Let $v\notin\mathbf T$ be a critical value of $p$ and $w$ the other one. Choose a simple arc $\gamma$ from $v$ to a point $\zeta\in\mathbf T$, $\zeta\ne w$, avoiding $w$ and meeting $\mathbf T$ only at $\zeta$. Such an arc exists in either component of $\mathbb C\setminus\mathbf T$, with one point removed if necessary. The set of $t\in\gamma\setminus\{\zeta\}$ for which some maximizer has critical values $t$ and $w$ contains $v$ and is relatively open by the deformation.

It is also relatively closed. Center each maximizer by translation, writing it as $z^3-3u_j^2z+b_j$. If its critical values are $t_j\to t$ and $w$, then $b_j=(t_j+w)/2$ and $|u_j|^3=|w-t_j|/4$ are bounded. A subsequence converges to a monic cubic with critical values $t$ and $w$, and its length is maximal by Lemma 4. The connectedness of $\gamma\setminus\{\zeta\}$ therefore gives every $t$ on that arc. The same compactness argument at $\zeta$ gives a maximizer with the two distinct critical values $\zeta$ and $w$.

If $w\notin\mathbf T$, repeat with an arc from $w$ to a point of $\mathbf T\setminus\{\zeta\}$ avoiding $\zeta$. This produces a maximizer with two distinct critical values on $\mathbf T$, which the strict family inequality excludes.

Therefore every maximizer is $(z-z_0)^3-w$ with $|w|=1$. Conversely each such polynomial is a rotation and translation of $z^3-1$ and has length $L_0$.